2011/045) What is the remainder when x^81 + x^49 + x^25 + x^9 + x is divided by x^3 - 1

when we divide by x^3-1 it shall be a maximal quadratic polynomial

ax^2 + bx + c

x^81 - 1 is divisible by x^3 -1 so x^81 divided by x^3-1 leaves 1

x^48 -1 divided by x^3 -1 leaves 0

so x(x^48-1) divided by x^3-1 leaves 0
x^49 divided by x^3-1 leaves x

x^25 divided by x^3-1 leaves x( similarly)
x^9 divided by x^3-1 leaves 1
so adding all the above we get remainder 1 + x + x + 1+ x = 3x + 2